Problem 22 - Precalculus Essentials

Question

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$y^{2}=x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the equation \(y^{2} = x^{2} - 2\) is symmetric with respect to the y-axis, the x-axis and the origin.

Step 1: Checking Symmetry with respect to the Y-Axis

To check symmetry with respect to the y-axis, replace \(x\) in the original equation with \(-x\). The original equation is \(y^{2} = x^{2} - 2\). If we replace \(x\) with \(-x\), we get \(y^{2} = (-x)^{2} - 2\), which simplifies to \(y^{2} = x^{2} - 2\), the original equation. So it is symmetric with respect to the y-axis.

Step 2: Checking Symmetry with respect to the X-Axis

To check symmetry with respect to the x-axis, replace \(y\) in the original equation with \(-y\). If we replace \(y\) with \(-y\), we get \((-y)^{2} = x^{2} - 2\), which simplifies to \(y^{2} = x^{2} - 2\), the original equation. So it is symmetric with respect to the x-axis.

Step 3: Checking Symmetry with respect to the Origin

To check symmetry with respect to the origin, both \(x\) and \(y\) in the original equation should be replaced with their negatives. If \(-x\) and \(-y\) are both replaced in the original equation, we get \((-y)^{2} = (-x)^{2} - 2\), which simplifies to \(y^{2} = x^{2} - 2\), the original equation. So it is symmetric with respect to the origin.